Existence and Multiplicity of Solutions for a Class of Kirchhoff–Boussinesq-Type Problems with Logarithmic Growth

In this paper, two problems related to the following class of elliptic Kirchhoff–Boussinesq-type models are analyzed in the subcritical ( \(\beta =0\) $\beta =0$ ) and critical ( \(\beta =1\) $\beta =1$ ) cases: \(\begin{aligned} \Delta ^{2} u \!- \!\Delta _p u \!=\! \tau |u|^{q-2} u{\ln |u|}\!+\!\beta |u|^{2_{**}-2}u\ \text{ in } \ \Omega \ \ \text{ and } \ {\Delta u=u=0} \ \text{ on } \ \ \partial \Omega , \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}^{2}u-{\mathrm{\Delta }}_{p}u={\tau |u|}^{q-2}uln|u|+\beta {|u|}^{2_{\ast \ast }-2}u\text{in}\mathrm{\Omega }\text{and}\mathrm{\Delta }u=u=0\text{on}\partial \mathrm{\Omega },\end{array}$$ where \(\tau >0\) $\tau >0$ , \(2< p< 2^{*}= \frac{2N}{N-2}\) $2<p<2^{\ast }=\frac{2N}{N-2}$ for \( N\ge 3\) $N\ge 3$ and \(2_{**}= \infty \) $2_{\ast \ast }=\infty$ for \(N=3\) $N=3$ , \(N=4\) $N=4$ , \(2_{**}= \frac{2N}{N-4}\) $2_{\ast \ast }=\frac{2N}{N-4}$ for \(N\ge 5\) $N\ge 5$ . The first one is concerned with the existence of a nontrivial ground-state solution via variational methods. As for the second problem, we prove the multiplicity of such a solution using the Mountain Pass Theorem.